Chapter 12

For each example, how would I get the first image to look like the second?

What are these examples of?

A transformation of a geometric figure is a change in its position, shape, or size.

Types of transformations: reflection (flip), translation (slide), rotation (turn), dilation (shrink or grow)

Preimage – original figure before the transformation

Image – resulting figure after the transformation

An isometry is a transformation in which the preimage and image are congruent.

In other words, there is a change in position, but not shape or size.

A reflection is an isometry in which the orientation of the object and its image are opposites.

A reflection is an isometry in which the orientation of the object and its image are opposites.

ABCD is an image of KLMN. What is the image of angle L? Which side corresponds to NK?

Sometimes images are named as A’B’C’D’ with the ‘ (prime) signifying the difference between the image and pre-image.

∆XYZ has vertices X(-2,3), Y(1,1), and Z(2,4). Draw ∆XYZ and its reflection image in the x-axis. Name using primes.

∆XYZ has vertices X(-2,3), Y(1,1), and Z(2,4). Draw ∆XYZ and its reflection image in the line x=3. Name using new letters.

A translation is an isometry that maps all points of a figure the same distance in the same direction.

We describe translations using vectors <x,y>

Find the image of F under the translation<-4,1>.

2

1

-1

-2

-2 2 4

J

I

H

G

F

Find the vector that describes the translation H→I.

2

1

-1

-2

-2 2 4

J

I

H

G

F

Find the vector that describes the translation ∆ABC→ ∆A’B’C’.

4

3

2

1

-1

-2

-3

-4

-6 -4 -2 2 4 6

A'

C'

B'

C

B

A

Draw the image of ∆ABC under the translation <5,-2>.

To describe a rotation, you need three pieces of information:1. center of rotation (a point on or off the figure)

ON

Off

2. angle of rotation (positive number, 360 max.)

3. direction of rotation (clockwise or counterclockwise)

Draw the image that results when ABC is rotated counterclockwise 270° around the origin.

A composition of reflections in two parallel lines is a translation. two intersecting lines is a rotation.

A glide reflection is the composition of a glide (translation) and a reflection in a line parallel to the glide vector.

A figure has symmetry if there is an isometry that maps the figure onto itself.

Three types of symmetry: Line symmetry (a.k.a. reflectional

symmetry) Rotational symmetry – is its own image for

some rotation that is less than or equal to 180°

Point symmetry – has rotational symmetry of exactly 180°

What kind of symmetry does each figure have? (could be multiple types)

A tessellation is a repeating pattern of figures that completely covers a plane, without gaps or overlaps.

All triangles and quadrilaterals tessellate.

A regular polygon will tessellate a plane if the interior angle measure will divide into 360 evenly.

A dilation is a transformation whose preimage and image are similar. It is generally not an isometry.

Every dilation has a center and a scale factor. The scale factor describes the size change

from the original figure to the image.

The dilation is an enlargement if the scale factor n > 1.

It is a reduction if the scale factor 0 < n < 1.

The green circle is a dilation of the blue circle. Describe the dilation.

3 cm

8 cm

C

∆ABC is a dilation of ∆DBC. Find the center and scale factor.

2 in.

6 in.

E

A

B

C

D

The scale factor on a museum's floor plan is 1 : 200. The length and width on the drawing are 8 in. and 6 in. Find the actual dimensions in feet and inches.

∆XYZ has coordinates X(3,1), Y(2,-4), and Z (-2,0). Find the image for a dilation with center (0,0) and scale factor 2.5.